When dividing terms with the same bases but different exponents, you will need to subtract all the pertinent exponents.

\(\displaystyle \frac {x^9}{x^3}\) becomes \(\displaystyle x^6\),

\(\displaystyle \frac {y^5}{y^2}\) becomes \(\displaystyle y^3\),

and \(\displaystyle \frac {1}{z}\) stays the same because there is no other z term to combine with it.

Thus resulting in:

\(\displaystyle \frac{x^6y^3}{z}\)

Begin by noting that the group (4²³ - 4²¹) has a common factor, namely 4²¹.  You can treat this like any other constant or variable and factor it out.  That would give you: 4²¹(4² - 1). Therefore, we know that:

3² * (4²³ - 4²¹) = 3² * 4²¹(4² - 1)

Now, 4² - 1 = 16 - 1 = 15 = 5 * 3.  Replace that in the original:

3² * 4²¹(4² - 1) = 3² * 4²¹(3 * 5)

Combining multiples withe same base, you get:

3³ * 4²¹ * 5

The first step in the problem is to combine like terms in the numerator, remembering that \(\displaystyle a^{m}\cdot a^{n} = a^{m+n}\):

\(\displaystyle \frac{a^{3}y^{3}y^{4}z^{-2}}{a^{3}y^{2}z^3} = \frac{a^{3}y^{7}z^{-2}}{a^{3}y^{2}z^3}\)

Next, we resolve the numerator, using \(\displaystyle \frac{a^{m}}{a^{n}}=a^{m-n}\) and \(\displaystyle a^{0} =1\): 

\(\displaystyle \frac{a^{3}y^{7}z^{-2}}{a^{3}y^{2}z^3} = y^{5}z^{-5}\)

Lastly, simplify the negative exponent using \(\displaystyle a^{-m}=\frac{1}{a^{m}}\)

 \(\displaystyle y^{5}z^{-5} = \frac{y^{5}}{z^{5}}\)

 Thus,  

\(\displaystyle \frac{a^{3}y^{3}y^{4}z^{-2}}{a^{3}y^{2}z^3} = \frac{y^{5}}{z^{5}}\)

The first step is to simplify each fraction by dividing like terms, remembering that \(\displaystyle \frac{a^{m}}{a^{n}}=a^{m-n}\), to get: 

\(\displaystyle \frac{wx^{3}y^{2}z^{-3}}{w^{2}x^{3}z}\cdot \frac{w^{-2}x^{3}}{x^{3}z^{-5}} = (w^{-1}y^{2}z^{-4})\cdot (w^{-2}z^{5})\)

Next, combine using multiplication and the rule \(\displaystyle a^{m}\cdot a^{n}= a^{m+n}\):

\(\displaystyle (w^{-1}y^{2}z^{-4})\cdot (w^{-2}z^{5}) = w^{-3}y^{2}z\)

Since the problem specifies that we must avoid fractions, we will not eliminate the negative exponents.

So, 

\(\displaystyle \frac{wx^{3}y^{2}z^{-3}}{w^{2}x^{3}z}\cdot \frac{w^{-2}x^{3}}{x^{3}z^{-5}} = w^{-3}y^{2}z\)

When dividing exponential expressions with the same base, subtract the exponents. In this problem, the exponents are \(\displaystyle 10\) and \(\displaystyle 6\). When subtracted, the result is \(\displaystyle 4\). Thus, the correct answer is \(\displaystyle x^{4}\).

A is not equivalent because exponents in denominators mean subtraction of exponents and not division of them. Furthermore, A, when computed, comes out to \(\displaystyle 1000\) instead of \(\displaystyle 10000\).

B is equivalent by the aforementioned exponential property, while C is simply computing the expression.

When two exponents with the same base are being divided, subtract the exponent of the denominator from the exponent of the numerator to yield a new exponent. Attach that exponent to the base, and that is your answer. 

In this case, subtract \(\displaystyle 1\) from \(\displaystyle 3\). That yields \(\displaystyle 2\) as the new exponent and \(\displaystyle x^{2}\) as the answer.